If you don't check that the translation is good,
then there is every chance that either:

the proofs you carry out
in your rich system do not correspond
naturally to proofs in Z_2;

your rich system does not share the same consequence
relation as Z_2;

(btw I use SOA and Z_2
interchangeably to mean `a reasonable axiomatisation
of the usual SOA consequence relation').

I can't say what the best books are, since I picked up
most of what I know from fairly obscure sources. The text
I have read that gives the best feel of what life is like in a
a proof theory based on the lambda calculus is Jean-Yves
Girard's `Proofs and Types' (with appendices by Paul Taylor
and Yves Lafont). This text gives many of the most important
results, in particular the proof of strong normalisation for
System F and Martin-Loef's demonstration that this SN
result is equivalent to the consistency of SOA, but be warned
that this text is challenging, cavalier about definitions and
has a very non-mainstream agenda. Also the book does not
explain why polymorphism is not set-theoretic; for this I
recommend John Reynold's orginal paper.